ap3290_chapter_4_i_10

Thermodynamics Y Y Shan

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Chapter 4 Thermodynamic Entropy

Introduction: In science, the term "entropy", which means transformation in Greek, has two

distinct interpretations: one is the thermodynamic entropy in the field of physics, the other is information entropy in information theory. Thermodynamic entropy, being the subject to be discussed in the following, is a fundamentally different concept from information entropy in information theory

There are two viewpoints explaining the thermodynamic entropy. Classical thermodynamics gives a macroscopic viewpoint, while statistical thermodynamics gives a microscopic viewpoint. From the two different viewpoints, two equivalent definitions of entropy had been developed in classical thermodynamics and statistical thermodynamics, respectively.

The definition of entropy in classical thermodynamics was developed in the early 1850s by thermodynamicist Rudolf Clausius. It makes no reference to the microscopic nature of matter. The entropy is interpreted simply as a state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved.

The definition of entropy in statistical thermodynamics was developed by Ludwig Boltzmann in the 1870s. This definition describes the entropy as a measure of the number of possible microscopic configurations of the individual atoms and molecules of the system (microstates)

1. Macroscopic perspective of entropy in classical thermodynamics: a State Function

In thermodynamics, if the balance of function Z remains unchanged during a cyclic process, i.e.,


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= 0dZ ,

or the difference between the initial state and final state is independent of paths, i.e.,

iffi ZZZ =

Then function Z is called a state function, such as the internal energy U, 0=U .

1.1 An example to calculate Q/T:

In the following thermodynamic cycle, if a function is defined as: TQS = , so that

its infinitesimal change: TdQdS = .

What is the change of the function after a cycle ABCDA:

=== ?TdQdSS ABCDA


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==

==

==

==

+++=

A

D

DADA

D

C

CDCD

C

B

BCBC

B

A

ABAB

DACDBCABABCDA

TdQS

TdQS

TdQS

TdQS

SSSSS

Therefore, the function S as defined is a state function

1.2. Clausius theorem: Entropy a state function proved mathematically

It has been obtained in chapter 2, that for a Carnot engine cycle, the efficiency is :

C

C

H

H

H

C

H

C

H

C

H

C

CCarnot

TQ

TQ

TT

QQ

TT

QQ

QW

==

===

thatso

,11

From above, we get the even more useful relation:


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0=C

C

H

H

TQ

TQ

eqn4-3

If we take Q to be positive value, representing heat energy added to the system, and take Q to be negative value, representing heat taken from the system, 4-3 can be written as:

CH

C

C

H

H

TTTTTQ

TQ

TQ

TQ

===+

=+

212

2

1

1 , e wher , 0

or ,0

In principle, any loop(cycle) can be broken up into a grid of infinitesimal Carnot cycles, as shown in the following:.

The smooth closed curve(black) is a reversible cycle and the zigzag closed path is made up of alternating isothermal(red) and adiabatic(blue) processes from neighboring Carnot cycles.

For the far left Carnot cycle abcd, we have


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0 2

2

1

1=+

TQ

TQ

Then for the next Carnot cycle efgh,

0 4

4

3

3=+

TQ

TQ

Similar equations can be written for each Carnot cycle and if all the equations are added, then the result obtained is that

0.... 5

5

4

4

3

3

2

2

1

1=+++++

TQ

TQ

TQ

TQ

TQ

Since no heat is transferred during the adiabatic portions of the zigzag cycle, we may write

0=j j

j

TQ

where the summation is taken over the entire zigzag cycle consisting of Carnot cycles, j in number. Now, imagine the cycle divided into a very large number of strips of very narrow Carnot cycles. When the isothermal processes (ab, ef) become shorter, then a zigzag path can be made to approximate the original cycle as closely as we please. When these isothermal processes become infinitesimal, in limit we obtain:

== 0 0 TdQ

TQ

j j

j 4.5

It is called the Clausius theorem. This result proves that the function defined as

is a state function. This function S, namely Entropy (in Greek, meaning "transformation"), is developed in the 1850s by German physicist Rudolf Clausius. This is often a sufficient definition of entropy if you don't need to know about the microscopic details. It can be integrated to calculate the change in entropy S during any part of a thermodynamic cycle.


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==B

AABBA T

dQSSS

For the case of an isothermal process(constant temperature) it can be evaluated simply by S = Q/T. The unit for entropy is Joules/Kelvin (J/K) or eV/K.

1.3 An entropy generator: Joule paddle wheel apparatus It is instructive to calculate a unit of entropy, J/K, in order to gain a feeling for this new

thermodynamic variable. Consider the Joule paddle wheel apparatus shown in the following. The system is 1kg water at T=300K T.

The surroundings are the adiabatic cylindrical wall, the diathermic bottom in contact with a heat reservoir, also at T=300K, and the paddle wheels.

A slowly falling mass m causes the paddle wheels to turn. The falling mass does work on the system, which causes an increase in the temperature of the water. However, the

diathermic bottom prevents the temperature from rising by removing heat energy Q from the system into the reservoir.

Entropy generator

The change of entropy of the reservoir is given by:

TQ

TdQS

reservoir ==

Since the state of the water (temperature and volume) is unchanged at the end of the process, the internal energy is unchanged.


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00

===

QWUS

water

water

Thus, the work W, done by the falling mass, equals the heat Q that enters the reservoir. If the paddle wheel is driven by a mass of 29.9 kg that falls 1 meter,

mghWQ == The entropy of the mass turning the paddle wheels is not changed during the process, because no heat enters or leaves the mass.

0=mass

S Then the change of entropy is given by:

)/(0.1300

1/8.99.29 KJK

mkgNkgTQSSSS

masswaterreservoir =

==++=

The entropy is generated by the conversion of work (done by the mass) into heat (entering the reservoir). Thus, the paddle wheel apparatus serves as an entropy generator.

1.4 The entropy version of the second law : principle of entropy increasing:

There are different versions, but all equivalant to each other, of the second law of thermodynamics. In chapter 3, the second law in its heat engine formulation, by Lord Kelvin, is:

It is impossible to convert heat completely into work. And in its heat formulation by Clausius is:

Heat cannot spontaneously flow from a material at lower temperature to

a material at higher temperature. The version of the second law that refers to entropy directly is due to Rudolf Clausius

In an isolated thermodynamic system, a process can occur only if it increases the total entropy of the system

Thus, the thermodynamics system can either stay the same, or undergo some physical process that increases its entropy.

Therefore, we can say: the entropy of the universe tends to increase, this the principle of entropy increasing. Here the universe is defined as:


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universe(isolated thermodynamic system) = a system + its surroundings

0)( universeS

Ice melting - a classic example of entropy increasing described in 1862 by Rudolf Clausius

A small 'universe', consists of the 'surroundings' (the hot room) and 'system' (glass, ice, cold water). In this universe, some heat energy dQ from the warmer surroundings(at 300K) will spread out to the cooler system of ice and water at its T of 273 K , the melting temperature of ice. The heat dQ for this process is the energy required to change water from the solid state to the liquid state. The entropy of the system will change by the amount:

KdQ

TdQdS

icesystem 273

==

The entropy of the surroundings will change by an amount:

KdQ

TdQdS

room

gsurroundin 300

=

=

So in this example, the entropy of the system increases, whereas the entropy of the

surroundings decreases.

ap3290_chapter_4_i_10

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